Lecture Note for Bounded Controls in Continuous-Time and Control of Several Variables

TL;DR

This paper develops a first-order optimal control theory for problems with box constraints, detailing modifications to Pontryagin's maximum principle and implementation methods for scalar quadratic Hamiltonians.

Contribution

It introduces the first comprehensive theory for bounded controls in continuous-time optimal control, including projection formulas and forward-backward sweep methods.

Findings

Modified Pontryagin's maximum principle for box constraints

Projection/clamping formula for scalar quadratic Hamiltonians

Implementation via forward-backward sweep method

Abstract

In this note, we develop the first-order theory of optimal control problems with box constraints on the control. We emphasize the precise modification of Pontryagin's maximum principle when the admissible control set is compact, the projection/clamping formula for scalar quadratic Hamiltonians, the distinction between intrinsic projection inside the optimality system and post hoc truncation of an unconstrained solution, and the corresponding forward-backward sweep implementation. The presentation is pitched at senior PhD students who are already comfortable with variational arguments, adjoint systems, and basic nonlinear analysis. These notes are mainly based on the book ``optimal control applied to biological models'' of Suzanne Lenhart and John T. Workman.